Mapping - Revision history

BrianMcMahon: Tidied translations.

2017-12-11T09:08:18Z

Tidied translations.

BrianMcMahon: Tidied translations and added German and Spanish (U. Mueller)

2017-11-16T12:37:54Z

Tidied translations and added German and Spanish (U. Mueller)

BrianMcMahon at 09:21, 21 May 2017

2017-05-21T09:21:51Z

BrianMcMahon: Style edits to align with printed edition

2017-05-15T16:21:19Z

Style edits to align with printed edition

MassimoNespolo: Lang

2015-03-04T05:49:30Z

Lang

BerndSouvignier at 13:39, 1 April 2009

2009-04-01T13:39:59Z

MassimoNespolo at 18:48, 12 March 2009

2009-03-12T18:48:38Z

New page

The term '''mapping''' is often used in mathematics as a synonym of ''function''. In crystallography it is particularly used to indicate a ''transformation''.

==Domain, image and codomain==
[[Image:Domain.jpg|thumb|200px|Definition of domain, image and codomain]] A mapping ''f'' of ''X'' to ''Y'' (''f'' : ''X'' → ''Y'') acts on a '''domain''' ''X'' and the result is defined in the '''codomain''' ''Y'' of the function.

The set of values ''f''(''X'') is the '''image''' of the function. The image may span the whole codomain or be a subset of it.

==Surjective, injective and bijective mappings==
[[Image:Surjection.jpg|thumb|200px|In a surjective mapping the image spans the whole codomain]] The mapping ''f'' is '''surjective''' ('''onto''') if the image coincides with the codomain. The mapping is '''many-to-one''' because more than one element of the domain ''X'' can be mapped to the same element of the codomain ''Y'', but all elements of ''Y'' are mapped to some elements of ''X''. A surjective mapping is a '''surjection'''.

[[Image:Injection.jpg|thumb|200px|An injective mapping is a one-to-one mapping]] The mapping ''f'' is '''injective''' ('''one-to-one mapping''') if different elements of the domain ''X'' are mapped to different elements in the codomain ''Y''. The image does not have to coincide with the codomain and therefore there may be elements of ''Y'' that are not mapped to some elements of ''X''. An injective mapping is an '''injection'''.

[[Image:Bijection.jpg|thumb|200px|A bijective mapping is a one-to-one correspondence]] The mapping ''f'' is '''bijective''' ('''one-to-one correspondence''') if and only if it is both injective and surjective. Every element of the codomain ''Y'' is mapped to exactly one element of the domain ''X''. The image coincides with the codomain. A bijective function is a '''bijection'''.

[[Category:Fundamental crystallography]]

← Older revision		Revision as of 09:08, 11 December 2017
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−	<font color="blue">Transformation </font>(''Fr''). <font color="red">Abbildung</font> (''Ge''). <font color="black">Trasformazione </font>(''It''). <font color="purple">写像</font>(''Ja''). <font color="green">Transformación geométrica</font> (''Sp'').	+	<font color="blue">Transformation</font> (''Fr''). <font color="red">Abbildung</font> (''Ge''). <font color="black">Trasformazione</font> (''It''). <font color="purple">写像</font> (''Ja''). <font color="green">Transformación geométrica</font> (''Sp'').

← Older revision		Revision as of 12:37, 16 November 2017
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−	<~~Font~~ color="blue">Transformation </~~Font~~>(''Fr''). <~~Font~~ color="black">Trasformazione </~~Font~~>(''It''). <~~Font~~ color="purple">写像</~~Font~~>(''Ja'').	+	<font color="blue">Transformation </font>(''Fr''). <font color="red">Abbildung</font> (''Ge''). <font color="black">Trasformazione </font>(''It''). <font color="purple">写像</font>(''Ja''). <font color="green">Transformación geométrica</font> (''Sp'').

← Older revision		Revision as of 09:21, 21 May 2017
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−	<Font color="blue">Transformation </Font>(''It''). <Font color="black">Trasformazione </Font>(''It''). <Font color="purple">写像</Font>(''Ja'').	+	<Font color="blue">Transformation </Font>(''Fr''). <Font color="black">Trasformazione </Font>(''It''). <Font color="purple">写像</Font>(''Ja'').

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-<Font color="blue">Transformation </Font>(''It''). <Font color="black">Trasformazione </Font>(''It''). <Font color="purple">写像</Font>(''Ja'')
+<Font color="blue">Transformation </Font>(''It''). <Font color="black">Trasformazione </Font>(''It''). <Font color="purple">写像</Font>(''Ja'').
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 ==Domain, image and codomain==
-[[Image:Domain.jpg|thumb|200px|Definition of domain, image and codomain]] A '''mapping''' ''f'' of ''X'' to ''Y'' (''f'' : ''X'' &rarr; ''Y'') assigns to each element ''x'' in the '''domain''' ''X'' a value ''y'' in the '''codomain''' ''Y''.
+[[Image:Domain.jpg|thumb|200px|Definition of domain, image and codomain.]] A '''mapping''' ''f'' of ''X'' to ''Y'' (''f'' : ''X'' &rarr; ''Y'') assigns to each element ''x'' in the '''domain''' ''X'' a value ''y'' in the '''codomain''' ''Y''.
-The set of values ''f(X)'' = { ''f(x)'' : ''x'' in ''X'' } is the '''image''' of the mapping. The image may be the whole codomain or a proper subset of it.
+The set of values ''f''(''X'') = { ''f''(''x'') : ''x'' in ''X'' } is the '''image''' of the mapping. The image may be the whole codomain or a proper subset of it.
-<br>
-For an element ''y'' in the image of ''f'', the set { ''x'' in ''X'' : ''f(x) = y'' } of elements mapped to ''y'' is called the '''preimage''' of ''y'', denoted by ''f<sup> -1</sup>{y}''. Also, the single elements in ''f<sup>  -1</sup>{y}'' are
+For an element ''y'' in the image of ''f'', the set { ''x'' in ''X'' : ''f''(''x'') = ''y'' } of elements mapped to ''y'' is called the '''preimage''' of ''y'', denoted by ''f''<sup>&minus;1</sup>{''y''}. Also, the single elements in ''f''<sup>&minus;1</sup>{''y''} are
 called '''preimages''' of ''x''.
 ==Surjective, injective and bijective mappings==
-[[Image:Surjection.jpg|thumb|200px|In a surjective mapping the image is the whole codomain]] The mapping ''f'' is '''surjective''' ('''onto''') if the image coincides with the codomain. The mapping may be '''many-to-one''' because more than one element of the domain ''X'' can be mapped to the same element of the codomain ''Y'', but every element of ''Y'' has a preimage in ''X''. A surjective mapping is a '''surjection'''.
+[[Image:Surjection.jpg|thumb|200px|In a surjective mapping the image is the whole codomain.]] The mapping ''f'' is '''surjective''' ('''onto''') if the image coincides with the codomain. The mapping may be '''many-to-one''' because more than one element of the domain ''X'' can be mapped to the same element of the codomain ''Y'', but every element of ''Y'' has a preimage in ''X''. A surjective mapping is a '''surjection'''.
-[[Image:Bijection.jpg|thumb|200px|A bijective mapping is a one-to-one correspondence]] The mapping ''f'' is '''bijective''' ('''one-to-one correspondence''') if and only if it is both injective and surjective. Every element of the codomain ''Y'' has exactly one preimage in the domain ''X''. The image coincides with the codomain. A bijective mapping is a '''bijection'''.
+[[Image:Bijection.jpg|thumb|200px|A bijective mapping is a one-to-one correspondence.]] The mapping ''f'' is '''bijective''' ('''one-to-one correspondence''') if and only if it is both injective and surjective. Every element of the codomain ''Y'' has exactly one preimage in the domain ''X''. The image coincides with the codomain. A bijective mapping is a '''bijection'''.
 <br>
-If the codomain of an injective mapping ''f'' is restricted to the image ''f(X)'', the resulting mapping is a bijection from ''X'' to ''f(X)''.
+If the codomain of an injective mapping ''f'' is restricted to the image ''f''(''X''), the resulting mapping is a bijection from ''X'' to ''f''(''X'').
 [[Category:Fundamental crystallography]]

← Older revision		Revision as of 05:49, 4 March 2015
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		+	<Font color="blue">Transformation </Font>(''It''). <Font color="black">Trasformazione </Font>(''It''). <Font color="purple">写像</Font>(''Ja'')
		+
		+
	The term '''mapping''' is often used in mathematics as a synonym of ''function''. In crystallography it is particularly used to indicate a ''transformation''.		The term '''mapping''' is often used in mathematics as a synonym of ''function''. In crystallography it is particularly used to indicate a ''transformation''.